nLab immersion of smooth manifolds

Definition

Let $f \colon X \to Y$ be a differentiable function between smooth manifolds (or just differentiable manifolds) of finite dimension.

We denote by $T X$, $T Y$ the tangent bundles and by $(-)\times_Y (-)$ the fiber product of differentiable functions into $Y$. In particular, $f^\ast T Y \,=\, X \times_Y T Y$ is the pullback bundle of $T Y$ along $f$ to a real vector bundle over $X$.

Definition

The differentiable function $f : X \to Y$ is called an immersion precisely if the canonical map

$$T X \longrightarrow X \times_Y T Y \eqqcolon f^* T Y$$

is a monomorphism.

This map is the one induced from the universal property of the pullback by the commuting diagram

$$\array{ T X &\overset{d f}{\longrightarrow}& T Y \\ \big\downarrow && \big\downarrow \\ X &\underset{f}{\longrightarrow}& Y }$$

given by the differential of $f$ going between the tangent bundles.

Equivalently this means the following:

Definition

The function $f : X \to Y$ is an immersion precisely if for every point $x \in X$ the differential

$$d f|_x \;\colon\; T_x X \to T_{f(x)} Y$$

between the tangent space of $X$ at $x$ and the tangent space of $Y$ at $f(y)$ is an injection.

Definition

(embedding of smooth manifolds)

An immersion whose underlying continuous function is an embedding of topological spaces is called an an embedding of smooth manifolds.

Example

(immersions that are not embeddings)

Consider an immersion $f \;\colon\; (a,b) \to \mathbb{R}^2$ of an open interval into the Euclidean plane (or the 2-sphere) as shown on the right. This is not an embedding of smooth manifolds: around the points where the image crosses itself, the function is not even injective, but even at the points where it just touches itself, the pre-images under $f$ of open subsets of $\mathbb{R}^2$ do not exhaust the open subsets of $(a,b)$, hence do not yield the subspace topology.

Concretely, consider the function $\big($sin$(2-),$ sin$(-)\big) \;\colon\; (-\pi, \pi) \longrightarrow \mathbb{R}^2$. While this is an immersion and injective, it fails to be an embedding due to the points at $t = \pm \pi$ “touching” the point at $t = 0$.

figure from Lee (2012, Fig. 4.3)

Properties

Relation to embeddings

Immersion are precisely the “local embeddings”:

Proposition

A smooth map $f \colon X \to Y$ of smooth manifolds is an immersion precisely if for every point $x \in X$ there is an open neighbourhood $x \in U \subset X$ such that $f|_U \colon U \to Y$ is an embedding of smooth manifolds.

(e.g. Boothby, III Thm. 4.12)

Since moreover the embeddings of smooth manifolds admit local slice charts (see there) so do immersions.

In the generality of supermanifolds this is Varadarajan 2004, Thm. 4.4.3.

Relation to formal immersions

The related concept of formal immersion of smooth manifolds, defined as an injective bundle morphism $T M \to T N$ between tangent bundles, is in some ways easier to study, in the sense that the collection of all such formal immersions, $Imm^f(M, N)$, is simpler to analyze. Then under some conditions on $M$ and $N$ (see there), it is the case that the map $Imm(M, N) \to \Imm^f(M, N)$ is a weak homotopy equivalence. This is a case of the h-principle.

Characterization in differential cohesion

A smooth function $f : X \to Y$ between smooth manifolds is canonically regarded as a morphism in the cohesive (∞,1)-topos SynthDiff∞Grpd. With respect to the canonical infinitesimal neighbourhood inclusion $i :$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd there is a notion of formally unramified morphism in $SynthDiff\infty Grpd$.

$f$ is an immersion precisely if it is formally unramified with respect to this infinitesimal cohesion.

See the discussion at SynthDiff∞Grpd for details.

Variants

The algebraic geometry analogue of a submersion is a smooth morphism.

The analogue between arbitrary topological spaces (not manifolds) is simply an open map. There is also topological submersion, of which there are two versions.

Related concepts

• formal immersion of smooth manifolds

• submersion

• local diffeomorphism

• embedding of smooth manifolds

• isometric immersion

• regular value

References

• William M. Boothby, Def. III 4.3 in: An introduction to differentiable manifolds and Riemannian geometry, Academic Press (1975, 1986), Elsevier (2002) [ISBN:9780121160517, pdf]

• John Lee, Chapter 4 of: Introduction to Smooth Manifolds, Springer (2012) [doi:10.1007/978-1-4419-9982-5, book webpage, pdf]

Discussion in the generality of supermanifolds:

• Veeravalli Varadarajan, pp. 148 in: Supersymmetry for mathematicians: An introduction, Courant Lecture Notes in Mathematics 11, American Mathematical Society (2004) [doi:10.1090/cln/011]